= Upper-boundary qualification of mountain PV compression
A mountain does not by itself determine whether an atmospheric layer becomes thinner. In the fixed-lid model of <mountain-crossing isentropic PV balance>, raising the lower boundary by $h_m$ reduces layer depth from $H$ to $H-h_m$. Conserving <potential vorticity> then requires reduced absolute vertical <vorticity>; in the Northern Hemisphere this gives an anticyclonic relative tendency if other changes are neglected.
For an isentropic atmospheric layer, the relevant thickness is its <isentropic mass density>, not simply its geometric depth. Its upper <potential temperature> surface can rise, deform or exchange pressure loading as flow crosses terrain. Both boundaries and the thermodynamic structure must therefore be specified before inferring compression. The relation $f+\zeta_\theta=q_\theta\sigma_\theta$ gives the vorticity change once that mass-thickness change is known; it cannot determine the upper-boundary response on its own.
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